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Mirrors > Home > MPE Home > Th. List > latnlej1l | Structured version Visualization version GIF version |
Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.) |
Ref | Expression |
---|---|
latlej.b | ⊢ 𝐵 = (Base‘𝐾) |
latlej.l | ⊢ ≤ = (le‘𝐾) |
latlej.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
latnlej1l | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → 𝑋 ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latlej.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latlej.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | latlej.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | 1, 2, 3 | latnlej 17668 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍)) |
5 | 4 | simpld 495 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 lecple 16562 joincjn 17544 Latclat 17645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-lub 17574 df-join 17576 df-lat 17646 |
This theorem is referenced by: atnlej1 36397 3atlem4 36504 3atlem6 36506 dalemcnes 36668 lhpexle3lem 37029 cdlemd4 37219 cdlemd7 37222 cdleme0e 37235 cdleme3e 37250 cdleme9 37271 cdleme17c 37306 |
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